Specialized Hybrid Newton Schemes for Matrix pth Roots
Braulio De Abreu, Marlliny Monsalve, Marcos Raydan · 2008
We discuss different variants of Newton’s method for computing a pth root of a given matrix. A suitable implementation is presented for solving the Sylvester equation, that appears at every Newton’s iteration, via Kronecker products. This approach is quadratically convergent and stable, but too expensive in computational cost. In contrast we propose and analyze some specialized versions that exploit the commutation of the iterates with the given matrix. We establish convergence of all the new versions under mild assumptions on the initial guess, and stability for one of them. These versions are relatively inexpensive and can easily be combined, at the final stages, with the Newton-Kronecker implementation to avoid possible stability problems when high precision is required. Preliminary and encouraging numerical results of the hybrid schemes are presented for p = 3 and p =5 .